Max Planck Institut für Plasmaphysik

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1 ASDEX Upgrade Max Planck Institut für Plasmaphysik 2D Fluid Turbulence Florian Merz Seminar on Turbulence,

2 2D turbulence? strictly speaking, there are no two-dimensional flows in nature approximately 2D: soap films, stratified fluids, geophysical flows, magnetized plasmas

3 2D turbulence? a simplified situation (compared to 3D), more accessible to theoretical, experimental and computational approaches, interesting for developing and testing general ideas about turbulence much easier to visualize than 3D-turbulence interesting new phenomena (e.g. dual cascade)

4 Outline Basic equations Cascades in 2D Coherent structures

5 Basic equations: velocity 2D-Navier-Stokes equations for an incompressible fluid: D ( Dt v = t + v v = 0 ) v = 1 ρ p + f ext + ν 2 v where v is in the (x,y) plane, ( v ẑ = 0). For viscosity ν = 0 and f ext = 0 these equations are called Euler equations.

6 Basic equations: vorticity Taking the curl of the NS equations and discarding the zero x and y components of the equation gives ( D Dt ω = t + v ) ω = g + ν 2 ω for the vorticity ω = ( v ) ẑ. If g = ( f ext ) ẑ = 0 and ν = 0 (Euler equation), we have D Dt ω = 0 vorticity is conserved

7 Basic equations: energy and enstrophy Important quantities: mean energy E = 1 2 u2 de dt = 2νZ enstrophy Z = 1 2 ω2 (mean square vorticity) dz dt = 2ν ( ω) 2 in 2D with curl free forcing, energy and enstrophy can only decrease with time, they are conserved in the inviscid case (ν = 0) for ν 0 we get de dt 0, energy is a robust invariant in 2D dz dt does not necessarily go to zero for ν 0, enstrophy is a fragile invariant (dissipation anomaly!)

8 Dissipation anomaly as time evolves, vorticity patches get distorted by background velocity and generate smaller and smaller filaments the vorticity gradient increases and dz dt = 2ν ( ω)2 becomes sizeable dissipation even for ν 0: fragile invariant in the enstrophy cascade, the dissipation rate is independent of ν but depends only on the enstrophy transfer rate ν only determines the enstrophy dissipation scale, not the enstrophy dissipation rate

9 Exact results For forced, isotropic and homogenous turbulence: for small scales: Corrsin-Yaglom-relation for passive tracers D L (r) 2ν ds 2 dr = 4 3 ηr (D L (r) = (v L ( x) v L ( x + r))(ω( x) ω( x + r)) 2, S 2 (r) = (ω( x) ω( x + r)) 2 ), η = ν ( ω) 2 ) for large scales: 2D-analogue of the Kolmogorov relation (4/5-law) F 3 (r) = 3 2 ɛr (F 3 (r) = (v L ( x) v L ( x + r)) 3, ɛ = ν (ω) 2 = 2νZ) sign of the prefactor reversed!

10 Cascades

11 Direction of the energy/enstrophy transfer In spectral space, the expressions for energy and enstrophy read E = Z = E(k, t)dk k 2 E(k, t)dk... E 1 E 3 k 1 Z 1 k 2 Z 3 k 3... E 2 = E 1 + E 3, Z 2 = Z 1 + Z 3. E 2 Z 2 Energy and enstrophy conservation for three Fourier modes k 1, k 2 = 2k 1, k 3 = 3k 1

12 Direction of the energy/enstrophy transfer δe 1 + δe 2 + δe 3 = 0 k 2 1 δe 1 + k 2 2 δe 2 + k 2 3 δe 3 = 0 with δe i = E(k i, t 2 ) E(k i, t 1 ). Combining the equations gives δe 1 = 5 8 δe 2 δe 3 = 3 8 δe 2 k 2 1 δe 1 = 5 32 k2 2 δe 2 k 2 3 δe 3 = k2 2 δe 2 enstrophy goes to higher k (direct enstrophy cascade), energy goes to lower k (inverse energy cascade)

13 Direction of the energy/enstrophy transfer Alternatively: E = Z = E(k, t)dk k 2 E(k, t)dk an evolution of E(k, t) to larger k conflicts with the boundedness of enstrophy E(k, t) must evolve towards small wave numbers / large scales

14 KBL theory: dual cascade Kraichnan, Batchelor, Leith proposed the existence of a dual cascade in steady state turbulence ( 1968): energy/enstrophy is constantly injected at some intermediate k i direct enstrophy cascade to higher k dissipation scale k d = (β/ν 3 ) 1/6 (equivalent to Kolmogorov microscale in 3D) inverse energy cascade to lower k condenses in the lowest mode (for bounded domain) / is stopped by Ekman friction ( µ v-term in NS-equation) at (k E = µ 3 /ɛ) 1/2 E(k) is stationary, the transfer rates of energy ɛ and enstrophy β far from the dissipation scales are independent of k (self-similarity)

15 KBL theory: inertial ranges inertial range of the energy cascade: for k E k k i, the energy spectrum can only depend on ɛ. Dimensional analysis: k = [L] 1 ; E(k) = [L] 3 [T ] 2 ; ɛ = [L] 2 [T ] 3 E(k) = Cɛ 2/3 k 5/3 inertial range of the enstrophy cascade (k i k k d ): the energy spectrum can only depend on β. Dimensional analysis: k = [L] 1 ; E(k) = [L] 3 [T ] 2 ; β = [T ] 3 E(k) = C β 2/3 k 3 C, C constant and dimensionless. zero enstrophy transfer in the energy inertial range, zero energy transfer in the enstrophy inertial range

16 KBL: Inertial ranges in steady state turbulence

17 Experiments: soap films vertically flowing soap films are approximately 2D (thickness variations of about 10% - condition of incompressibility is slightly violated) turbulence is generated by grids/combs inserted in the flow

18 Experiments: soap films Energy spectrum in soap film

19 Correction to the energy spectrum the k 3 -spectrum of the enstrophy cascade gives rise to inconsistency: infrared divergence of for the (k-dependent) enstrophy transfer rate Λ(k) for k i 0 reason: contributions of larger structures to the shear on smaller structures (nonlocality in k-space). Kraichnan (TFM-closure approximation): logarithmic correction to restore constant transfer rate ( ) 1/3 k E(k) = C β 2/3 k 3 ln k i

20 Correction to the energy spectrum attempts to measure the corrections are being made: k 3 E(k) and (enstrophy flux)/β for various Reynolds numbers (direct numerical simulation [DNS] results)

21 Cascades: 2D vs. 3D Vorticity equation in 3 dimensions (analogous to MHD kinematic equation for ω B): D Dt ω = ( ω ) v + g + ν 2 ω the additional vortex-stretching term changes the behaviour significantly: gradients in the velocity field stretch embedded vortex tubes as the cross section decreases, the vorticity increases

22 Cascades: 2D vs. 3D enstrophy in 3D is not conserved even for ν = 0 but increases with time (in 2D: fragile invariant) no enstrophy cascade in 3D! energy in 3D follows de dt = 2νZ and is a fragile invariant (in 2D: robust invariant) energy cascades to smaller scales in 3D (direct cascade), in 2D to large scales (inverse cascade) E(k) has the same k 5/3 -dependence in the inertial range of the energy cascade

23 Cascades: 2D vs. 3D experimental results for grid turbulence in 3D/2D

24 Coherent structures

25 Coherent structures physical and numerical experiments: long lived vortical structures (lifetime turnover time) spontaneously emerging from the turbulent background these coherent structures alter the cascading behavior especially important in freely decaying turbulence (they can be inhibited / destroyed in forced systems) clear definition/identification of coherent structures difficult, several competing methods: e.g. simple threshold criteria, Weiss criterion, wavelet decomposition..

26 Coherent structures: axisymmetrisation experiments: elliptical structures are not stable but become circular exact theoretical result: circular patches of uniform vorticity are nonlinearly stable

27 Coherent structures: vortex merging experiments: if like sign vortices of comparable strength get too close, they merge (axisymmetrization) theory: analytical solution for the merging of two identical vortices

28 Coherent structures: vortex break-up experiments: the strain caused by the stronger vorices distorts weaker vortices up to destruction vorticity adds to background vorticity

29 Coherent structures: time evolution by vortex merging and break-up, the coherent structures in a freely decaying system become fewer and larger observables: evolution of vortex density ρ, typical radius a, intervortical distance r, extremal vorticity ω ext

30 Universal decay theory for dilute vortex gas empirical approach by Carnevale et al. assumption: two invariants E ρω 2 ext a4, contributions outside vortices negligible vorticity extremum ω ext of the system (observation) length scale l = E/ω ext, time scale τ = 1/ω ext Dimensional reasoning gives ρ = l 2 g(t/τ). Assumption g(t/τ) = (t/τ) ξ gives (ξ is to be measured) ρ l 2 (t/τ) ξ, a l(t/τ) ξ/4, Z τ 2 (t/τ) ξ/2 r l(t/τ) ξ/2, v E,

31 Universal decay theory : comparison with DNS left: decay law for the number of vortices right: inverse vortex density, intervortical distance, size, extremal vorticity (lines for ξ = 0.75)

32 Decay of vortex populations vortex merging and vortex break-up lead to ever larger and fewer coherent structures the system evolves towards a final dipole

33 Intermittency in 2D experimental results: no intermittency in 2D turbulence no theoretical explanation yet PDFs for longitudinal, transverse velocity increment (energy cascade) and the vorticity increment (enstrophy cascade) for several scales. δv = v L ( x) v L ( x+ r) δv = v T ( x) v T ( x+ r) δω = ω( x) ω( x+ r)

34 Intermittency in 2D hyperflatness H 2n (l) = F 2n(l) F 2 (l) n, F n (l) = δv (l) n (energy cascade) and structure functions of vorticity S n (l) = δω(l) n (enstrophy cascade) no intermittency, slight deviations from gaussianity are assumed to stem from coherent structures

35 Summary the existence of a dual enstrophy-energy cascade (KBL-theory) is experimentally confirmed coherent structures play an important role (especially in decaying turbulence) and modify the energy spectrum predicted by KBL-theory there is no intermittency found in experiments several systems of interest (e.g. geophysical flows, magnetized plasmas) are approximately 2-dimensional - results of 2D fluid turbulence are applicable

36 Further reading General 2D turbulence: P.A. Davidson, Turbulence, Oxford University Press (2004) M. Lesieur, Turbulence in Fluids, Kluwer (1997) U. Frisch, Turbulence, Cambridge University Press (1995) P. Tabelling, Two-dimensional turbulence: a physicist approach, Phys. Rep. 362, 1-62 (2002) Cascade classics Kraichnan, Inertial Ranges in Two-Dimensional Turbulence, Phys. Fluids 10, 1417 (1967) Leith, Diffusion Approximation for Two-Dimensional Turbulence, Phys. Fluids 11, 1612 (1968) Batchelor, Computation of the Energy Spectrum in Homogenous Two-Dimensional Turbulence, Phys. Fluids 12, II-233 (1969)